Abstract

The spread function and optical transfer function associated with image motion relative to the recording medium are derived and generalized to cover media whose response changes with time. Specific expressions are derived for an exponentially decaying response and three types of image motion: linear, sinusoidal, and random walk. An expression for the blurring in flying spot scanners using cathode ray tubes, caused by the finite decay time of the phosphor, is also derived.

© 1971 Optical Society of America

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References

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  1. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963), Chap. 5.
  2. L. Levi, Applied Optics (Wiley, New York, 1968), Sec. 3.2.
  3. A. Lohmann, Opt. Acta 6, 319 (1959).
    [CrossRef]
  4. R. M. Scott, Phot. Sci. Eng. 3, 201 (1959).
  5. D. P. Paris, Phot. Sci. Eng. 6, 55 (1962).
  6. Reference 2, Table 53.
  7. S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943), Chap. 1, Sec. 4a.
    [CrossRef]

1962 (1)

D. P. Paris, Phot. Sci. Eng. 6, 55 (1962).

1959 (2)

A. Lohmann, Opt. Acta 6, 319 (1959).
[CrossRef]

R. M. Scott, Phot. Sci. Eng. 3, 201 (1959).

1943 (1)

S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943), Chap. 1, Sec. 4a.
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943), Chap. 1, Sec. 4a.
[CrossRef]

Levi, L.

L. Levi, Applied Optics (Wiley, New York, 1968), Sec. 3.2.

Lohmann, A.

A. Lohmann, Opt. Acta 6, 319 (1959).
[CrossRef]

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963), Chap. 5.

Paris, D. P.

D. P. Paris, Phot. Sci. Eng. 6, 55 (1962).

Scott, R. M.

R. M. Scott, Phot. Sci. Eng. 3, 201 (1959).

Opt. Acta (1)

A. Lohmann, Opt. Acta 6, 319 (1959).
[CrossRef]

Phot. Sci. Eng. (2)

R. M. Scott, Phot. Sci. Eng. 3, 201 (1959).

D. P. Paris, Phot. Sci. Eng. 6, 55 (1962).

Rev. Mod. Phys. (1)

S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943), Chap. 1, Sec. 4a.
[CrossRef]

Other (3)

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963), Chap. 5.

L. Levi, Applied Optics (Wiley, New York, 1968), Sec. 3.2.

Reference 2, Table 53.

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Equations (37)

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δ ( x - x ) E 0 ( t ) d t ,
W ( x ; t f ) = t i t δ [ x - x ( t ) ] E 0 ( t ) H ( t f - t ) d t ,
δ [ x ( t ) ] = δ ( t - t 0 ) / | d x d t | ,
W ( x ; t f ) = t i t f δ ( t - t x ) v ( t ) E 0 ( t ) H ( t f - t ) d t = E 0 ( t x ) H ( t 5 - t x ) v ( t x ) ,
v = d x d t = [ ( d x / d t ) 2 + ( d y / d t ) 2 ] 1 2
T ( ν ) = - e i 2 π ν · x [ t i t f [ x - x ( t ) ] E 0 ( t ) H ( t f - t ) d t ] × d x / T ( o ) .
T ( ν ) = t i t f E 0 ( t ) H ( t f - t ) { - δ [ x - x ( t ) ] e i 2 π ν · x d x } × d t / T ( o ) = t i t f E 0 ( t ) H ( t f - t ) e i 2 π ν · x ( t ) d t / t i t f E 0 ( t ) H ( t f - t ) d t .
H ( t ) = e - α t .
t i = - ( τ / 2 ) , t f = τ / 2.
T ( o ) = - τ / 2 τ / 2 e - α ( τ / 2 - t ) d t = 2 α e - α τ / 2 sinh ( α τ / 2 ) ,
T ( ν ) = α 2 sinh ( α τ / 2 ) - τ / 2 τ / 2 e α t e i 2 π ν x ( t ) d t .
T ( ν ) = [ α cos w t + 2 w sin w t coth α τ 2 + i ( α sin w τ coth α τ 2 - 2 w τ ) ] α ( a 2 + 4 w 2 ) ,
T ( v ) 2 = 1 + sin 2 π ν v τ cosh ( 2 α τ / 2 ) 1 + ( 2 π ν v / a ) 2 ,
lim a 0 T ( v ) = sin π ν v τ π ν v T .
lim τ T ( ν ) = [ 1 + ( 2 π ν v / α ) 2 ] - 1 2 .
x = a sin w t + b cos w t
T ( v ) = - τ / 2 τ / 2 e α τ e i 2 π ν ( a sin ω t + b cos ω t ) d t / 2 α sinh α τ 2 .
T ( v ) = α n = - [ α J 2 n ( 2 π ν a ) α 2 + ( 2 n ω ) 2 + i ( 2 n + 1 ) ω J 2 n + 1 ( 2 π ν a ) α 2 + ( 2 n + 1 ) 2 ω 2 ] ,             α 0 = J 0 ( 2 π ν a ) , α = 0.
P ( x ) = c p ( x ) V ( x ) .
P ( x ) = c V e - r 2 / a 2 n t
L ( x ) = - P ( x 2 + y 2 ) 1 2 d y = c V e - x 2 / a 2 n t / ( π n t ) 1 2 a .
n = v / a ,
L ( x ) = [ c V / ( π v a t ) 1 2 ] e - x 2 / v a t .
T ( ν ) = exp ( - π 2 ν 2 v a t ) .
L ( x ) = c V ( π v a ) 1 2 0 τ 1 t 1 2 e - x 2 / v a t e - α t d t .
T ( ν ) = α 1 - e - α τ · 1 - exp [ - τ ( α + π 2 ν 2 a ) ] α + π 2 ν 2 v a .
T ( ν ) = x / α + π 2 ν 2 v a ,
T ( ν ) = 1 - exp ( - π 2 ν 2 v a τ ) / π 2 ν 2 v a τ .
x ( t ) = v t .
T ( ν ) = - τ / 2 τ / 2 e - α ( τ / 2 - t ) e i 2 π ν v t d t / 2 α sinh ( α τ 2 ) e - α τ / 2 .
T ( ν ) = α 2 sinh ( α τ / 2 ) - τ / 2 τ / 2 e a t ( cos β t + i sin β t ) d t = α 2 sinh ( α τ / 2 ) { 1 β e α t / β ( α 1 β ) 2 + 1 [ α β cos t + sin t + i × ( α β sin t - cos t ) ] } | - β τ / 2 β τ / 2 = α α 2 + β 2 [ α cos β τ 2 + β sin × β τ 2 coth α τ 2 + i ( α sin β τ 2 coth α τ 2 - β cos β t 2 ) ] T ( ν ) 2 = α 2 ( α 2 + β 2 ) 2 [ ( α cos β τ 2 + β sin β τ 2 coth α τ 2 ) 2 + ( α sin β τ 2 coth α τ 2 - β cos β τ 2 ) 2 ] = α 2 α 2 + β 2 × [ 1 + ( sin ( β τ / 2 ) sinh ( α τ / 2 ) ) 2 ] .
T ( ν ) = 1 2 α cosh α τ 2 I ,
I = - τ / 2 τ / 2 e α t [ cos ( A sin ω t ) + i sin ( A sin ω t ) ] d t = - τ / 2 τ / 2 e α t n = - [ J 2 n ( A ) cos 2 n ω t + i J 2 n + 1 ( A ) sin × ( 2 n + 1 ) ω t ] d t = n = - [ J 2 n ( A ) - τ / 2 τ / 2 e α τ cos 2 n ω t d t + i J 2 n + 1 ( A ) × - τ / 2 τ / 2 e α τ sin ( 2 n + 1 ) ω t d t = n = - { J 2 n ( A ) / 2 n ω ( α / 2 n ω ) 2 + 1 [ e α τ / 2 ( α 2 n ω cos n ω t + sin ω τ ) - e - α τ / 2 ( α 2 n ω cos n ω t - sin ω τ ) ] + i J 2 n + 1 ( A ) / ( 2 n + 1 ) ω [ ( α / ω ) / ( 2 n + 1 ) ] 2 + 1 × [ e α τ / 2 ( α / ω 2 n + 1 sin ( n + 1 2 ) ω τ - cos ( n + 1 2 ) ω τ ) + e - α τ / 2 ( α / ω 2 n + 1 sin ( n + 1 2 ) ω τ + cos ( n + 1 2 ) ω τ ) ] } .
I = 2 n = - [ α J 2 n ( A ) sinh α τ / 2 α 2 + ( 2 n ω ) 2 + i ( 2 n + 1 ) ω J 2 n + 1 ( A ) α 2 + ω 2 ( 2 n + 1 ) 2 × sinh α τ 2 ]
T ( ν ) = α n = - [ α J 2 n ( A ) α 2 + ( 2 n ω ) 2 + i ( 2 n + 1 ) ω J 2 n + 1 ( A ) α 2 + ( 2 n + 1 ) 2 ω 2 ] .
lim α I = τ J 0 ( A ) , lim α T ( ν ) = ( 1 / τ ) I = J 0 ( A ) .
T ( ν ) = c 0 τ e - π 2 ν 2 v a t e - α t d t c 0 τ e - k t d t = 1 k ( 1 - e - k t ) / 1 k ( 0 ) [ 1 - e - K ( 0 ) τ ] ,

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